Problem: The set of points $(x,y)$ such that $|x - 3| \le y \le 4 - |x - 1|$ defines a region in the $xy$-plane.  Compute the area of this region.
Answer: Plotting $y = |x - 3|$ and $y = 4 - |x - 1|,$ we find that the two graphs intersect at $(0,3)$ and $(4,1).$


[asy]
unitsize(1 cm);

real funcone (real x) {
  return(abs(x - 3));
}

real functwo (real x) {
  return(4 - abs(x - 1));
}

fill((3,0)--(4,1)--(1,4)--(0,3)--cycle,gray(0.7));
draw(graph(funcone,-0.5,4.5));
draw(graph(functwo,-0.5,4.5));
draw((-0.5,0)--(4.5,0));
draw((0,-0.5)--(0,4.5));

label("$y = |x - 3|$", (3.5,3));
label("$y = 4 - |x - 1|$", (0,1), UnFill);

dot("$(0,3)$", (0,3), W);
dot("$(4,1)$", (4,1), E);
dot("$(3,0)$", (3,0), S);
dot("$(1,4)$", (1,4), N);
[/asy]

The region then is a rectangle with side lengths $\sqrt{2}$ and $3 \sqrt{2},$ so its area is $(\sqrt{2})(3 \sqrt{2}) = \boxed{6}.$